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Journal of Physics B Atomic and Molecular Physics

Coherent propagation of two-photon quasi-resonant pulses in a medium of multi-level atoms

Farhad H. M. Faisal

Published Jul 11, 1977• 3 citations
Publisher

Abstract

Coherent propagation of a circularly polarized pulse which is two-photon quasi-resonant with a medium of multi-level atoms is considered in the adiabatic following limit. The relevant pair of non-linear coupled partial differential equations are solved exactly for arbitrary pulse shapes. The development of the envelope and the phase of the pulse as it propagates through the medium is described analytically. Perturbation theoretical results are shown to apply only for small intensity or small distance of propagation or both. Qualitatively different results apply for higher intensities and longer distances of propagation. These include strong self-phase-modulation effects as well as reshaping of the pulse envelope.

DOI
10.1088/0022-3700/10/10/029

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Journal of Physics B Atomic and Molecular Physics•May 28, 1980•1 citation
Coherent propagation of a circularly polarized pulse which is two-photon quasi-resonant with a medium of multi-level atoms is considered in the adiabatic following limit. The relevant pair of non-linear coupled partial differential equations are solved exactly for arbitrary pulse shapes. The development of the envelope and the phase of the pulse as it propagates through the medium is described analytically. Perturbation theoretical results are shown to apply only for small intensity or small distance of propagation or both. Qualitatively different results apply for higher intensities and longer distances of propagation. These include strong self-phase-modulation effects as well as reshaping of the pulse envelope.
Propagation of an ultrashort, small-area (circularly polarized) pulse through a gas medium of three-level atoms near one-photon resonance is studied. The expression for the propagating pulse shape is obtained within the rotating wave and dipole approximations. Results are compared, both analytically and numerically, with those for two-level model. Sinusoidal and Gaussian shapes of input pulses are compared, and a high sensitivity of modulation of the light pulse for envelope to its input shape is revealed.
The propagation of strongly polarized electromagnetic waves through a resonance medium, with arbitrary angular momenta of the levels, is investigated in the adiabatic-following approximation. Exact wavefunctions of the two-level atomic system with degenerate levels in the field of a circularly (linearly) polarized wave are constructed and the picture of the shift of energy level splittings is discussed. Exact formulae for the nonlinear refractive index of the medium in the field of the circularly (linearly) polarized wave are found. The propagation of counterpropagating polarized waves—an intense pump wave and a probe light signal—through the resonance medium is studied. The studies are carried out in the scope of a two-level (one-photon resonance) and a three-level (one- and two-photon resonances) medium. Formulae are found for the induced rotation of the plane of polarization for the probing signal that are exact in terms of the intensity of the pump wave. The spectral dependence of the angle of rotation is studied, and it is shown that a series of Stark-shifted resonance poles of atomic absorption and a three-photon scattering process arise.
Two-photon resonantly enhanced parametric generation processes have generally been described using time-dependent perturbation theory. In this paper we show that a theory of two-photon coherent effects can be used to derive and explain these nonlinear mixing processes. Our technique makes use of the adiabatic following (AF) approximation to obtain solutions to a vector model describing the two-photon resonance. We show that the usual results for the nonlinear susceptibilities correspond to the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ vector of Feynman, Vernon, and Hellwarth adiabatically following the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\gamma}}}$ vector in the small-angle limit. Consequently, the theory allows a natural extension to large angles, and power-dependent nonlinear susceptibilities are obtained. We then use these AF results for the polarization to study the propagation of pulses nearly resonant with a two-photon transition, and we demonstrate that the pulse reshaping is due to the two related effects of a nonlinear pulse velocity and self-phase modulation.
The authors develop a theoretical analysis which describes coherent nonlinear resonant effects where two optical pulses of different frequencies undergo two-photon self-induced transparency (two-photon SIT) in an inhomogeneously broadened three-level system. The field (Maxwell) and atomic (Schrodinger; density matrix representation) equations are coupled in a self-consistent manner by virtue of the resultant macroscopic polarizations at the two frequencies. This treatment is accomplished under circumstances where atomic coherence plays a predominant role. The resulting equations yield a new two-photon area theorem which gives first the intensity dependent, two-photon absorption law analogous to Beer's law for the one-photon transition. Secondly, a 2 pi Lorentzian pulse solution is derived which propagates without loss at the modified pulse velocity.
We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.
Evolution under two-photon near-resonance conditions of an adiabatic Gaussian pulse is considered in Rb vapour. Theoretical results show strong self-phase modulation and pulse deformation with increasing distance of propagation and intensity at a fixed detuning. At sufficiently large distances of propagation and/or input intensity pulse deformation is seen to develop into (two-photon) optical shocks. At a higher intensity double shock fronts are predicted to occur. The previously anticipated existence of an upper bound for the output maximum and of the vanishing of the self-phase modulation at the 'interaction time' between the output and the input profiles are shown explicitly to hold for the cases considered. A quasi free-wave like evolution occurs at asymptotically higher intensities.